TITLE: Sheaves and bundles in differential geometry
QUESTION [44 upvotes]: Because the theory of sheaves is a functorial theory, it has been adopted in algebraic geometry (both using the functor of points approach and the locally ringed space approach) as the "main theory" used to describe geometric data.  All sheaf data in the LRS approach can be described by bundles using the éspace étalé construction.  It's interesting to notice that the sheafification of a presheaf is the sheaf of sections of the associated éspace étalé.  
However, in differential geometry, bundles are for some reason preferred.  Is there any reason why this is true?  Are there some bundle constructions which don't have a realization as a sheaf?  Are there advantages to the bundle approach?

REPLY [3 votes]: In differential geometry one often also has connections on the bundles, e.g. the Levi-Cevita connection on the tangent bundle. Many concepts of differential geometry use connections, such as holonomy or geodesics. 
I'd love to learn the opposite to the following statement, but I have the feeling that there is no "nice" definition of a connection in the sheaf-theoretical approach. So I think this is another aspect why bundles are often preferred. 
As a little side remark, I'd like to point out that in higher differential geometry the story continues. Curiously, the word "gerbe" - which comes originally from the sheaf-theoretical side - is often at the same time used for the higher analog of a bundle. This is one of the reasons why different people associate so diverse meanings with "gerbe".